Golden Ratio from Fibonacci Sequence and Random Numbers; also SuperNacci Numbers

The Golden Ratio (also known as the Golden Mean  and the Divine Ratio), in my opinion is more significant that the otherwise known Most Important Number — Pi. Pi = 3.1415…… is the irrational number (the decimals do not terminate nor do they repeat) that is the ratio of the circumference of a circle to its diameter (perimeter divided by distance across circle through its center). The Golden Ratio is the relationship of all that is beautiful in nature (the children will explore this phenomenon the week of April 24-28). 

Adolf Zeising, whose main interests were mathematics and philosophy, found the golden ratio expressed in the arrangement of parts such as leaves and branches along the stems of plants and of veins in leaves. He extended his research to the skeletons of animals and the branchings of their veins and nerves, to the proportions of chemical compounds and the geometry of crystals, even to the use of proportion in artistic endeavors, and yes, human body proportions. In these patterns in nature he saw the golden ratio operating as a universal law. In 2010, the journal Science reported that the golden ratio is present at the atomic scale in the magnetic resonance of spins in cobalt niobate crystals. Since 1991, several researchers have proposed connections between the golden ratio and human genome DNA. The Golden Ratio can also be found in architecture, paintings, design, and music.

The Golden Ratio represented by the Greek letter phi which looks like a zero (0) with a vertical line (I) through its center is:

1.61803398875………

For example, if you took the height of any human and divided it by the height to that person’s belly button, you would get the Golden Ratio. 

I had the children use graphing calculators to key in any Fibonacci number in the list of the first 100. Then I had them divide that number by the previous Fibonacci number. I had them repeat this process several times without clearing their screen so they could see all of the quotients (answer to a division problem). They were astounded that most of the digits in each of their quotients were exactly the same: 1.61803398875.

Some of them were puzzled by the quotients of some very small Fibonacci numbers such as 1 divided by 1 = 1, 3 / 2 = 1.5 or 5 / 3 = 1.666, 8 / 5 = 1.6. Why were these numbers so different? Specifically, why did their calculators all display “Error: Divide by Zero” when they divided 1 / 0? Dividing by zero is essentially infinite so therefore, undefined. The higher you go in the Fibonacci sequence, the closer you come to the actual Golden Ratio. How could such an important number be generated by just a 0 and 1? 

Why not create the Golden Ratio with two random numbers? The children were skeptical so I had them choose two random numbers and treat them as the Fibonacci sequence: adding the previous two to get the next number. After every five calculations, I had them use the calculators to divide their largest number by the previous number. They were astounded to find that they were getting close to the Golden Ratio with their own sequence of numbers.

For the older children, I introduced the concept of Tribonacci numbers (0 0 1 1 2 4 7 13 24 …..) by adding the previous three numbers; Tetranacci numbers by adding the previous four numbers, Pentanacci numbers by adding the previous five numbers, and so on. These sequences also create a special ratio. They noticed that the higher you go in the ‘nacci numbers, the closer the ratio gets to the number 2 (but it can never reach 2, so 1.99999). Again, I had them explore these ratios by starting with random numbers. Yes, they still generate the same ratios.